Lets say we have an electron with known De Broglie wavelength $\lambda$. Can anyone justify or explain why we calculate its energy $E$ using 1st the De Broglie relation $\lambda = h/p$ to get momentum $p$ and 2nd using the invariant interval to calculate $E$:
\begin{align}
p^2c^2 &= E^2 – {E_0}^2\\
E &= \sqrt{p^2c^2 + {E_0}^2}
\end{align}
Why we are not alowed to do it like we do it for a photon:
\begin{align}
E=h\nu = h\frac{c}{\lambda}
\end{align}
These equations return different results.
Best Answer
Note that the equation $E=h\nu$ does not account for the energy equivalent of particle's mass. It assumes zero mass.
Photons has zero mass. You can actually substitute zero for $E_0$ so that $p^2c^2 = E^2$, and then apply de Broglie's relations so that $E = h\nu$.